A Bishop topology on \(X\)  is a certain subset of \(F̥(X)\) . As the Bishop topologies considered here
are all extensionalIf \(X\)  is a set and \(P\)  is an extensional property on \(X\)  i.e., \(P(x) \ \& \ x =_X y \Rightarrow P(y)\) ,
the extensional subset \(X_P\)  of \(X\)  is defined by separation, \(X_P = \lbrace x \in X \mid P(x)\rbrace \) , its equality is inherited by that
of \(X\)  and the embedding of \(X_P\)  into \(X\)  is defined by the identity rule (see [1]}, Definition 2.2.3).
subsets of \(F̥(X)\) , we do not mention the embedding \(i_F^{F̥(X)} \colon F \hookrightarrow F̥(X)\) ,
which is given in all cases by the identity map-rule. The uniform closure \(\overline{\Phi }\)  of \(\Phi \)  is
an extensional subset of \(F̥(X)\) .
